Rob Adams a Painter's Blog painter's progress

April 17, 2011

Spherical Perspective

Filed under: Drawing,How to do,Perspective — Tags: , — Rob Adams @ 11:30 am

This is the first instalment of a few posts on perspective and how we can use it in making images. This post is quite advanced but I do intend to do a “rule of thumb” perspective guide for those not of a mathematical bent. For now we are in the world of curved perspective which can be scary but explains a little about why even when we follow the construction rules of perspective things can look “wrong”. This is especially true when we use what in photographic terms is called a “wide angle”.



Spherical, Perspective, drawing

So here we go… We might assume from what we are taught about perspective that this is the way we actually see. But it’s not. In the outside world there are straight lines, so we put them that way into our pictures. We have developed complicated schemes of geometrical rules to guide us. We take photos with cameras that have lenses that carefully distort the world to make it fit with the expectation that straight line should be straight. But visually they are not. Have you ever tried to draw that really large checker board floor? Somehow at the far right and left it goes all stretched. Do the same thing with circles on the floor and it gets really wild. Just look at those ellipses on the far left they get really funky! Have you ever tried to stitch together that big panorama? They never quite fit do they. But if you take lots of pictures say every 5 degrees and just use the middle strip of each, it’s sort of easier. And when they’re all stuck together, well… those straight lines look distinctly curved. In camera terms we call this a Fish  Eye lens effect.


Spherical, Perspective, drawing

Spherical, Perspective, drawing

Let’s find out why this happens. Take the set up above. Simple enough, a railway track, a station and you. Now perspective and our eyes tell us that things get smaller as they move further away. I’ve no problem with that. So here goes. If we look straight ahead Say “B” then the track is quite close. If you drew it it would go straight across the page left to right. If we look to our right say “C” then the track in the center of our vision is a lot further away. On the right the track vanishes at a point on the horizon. These pictures are both fine but you wouldn’t want to try and join them together! But wait a second, the track really is joined together. And we haven’t moved. We just looked to our right…If we looked to out left then we’d see the track go to a point at the horizon again. So what have we got? one set of parallel lines and two that meet at points. This isn’t looking much like a railway track!


Spherical, perspective, drawing

Let’s see what’s really happening. Imagine if we do a whole set of tall narrow drawings turning our heads a little for each one and then stitch them together. This gives us the result above. If we were a chicken – or even a fish- this is actually how we might see it. We are not quite so different from them as you might think. You don’t, after all, keep your eyes still when you look at a scene. Indeed your eyes only do detailed looking with a small patch of our retina called the fovea. The brain then stitches all the bits together rather like you do in a photographic panorama. On top of all this we turn our heads. In real life we can soak up a huge vista of visual information and glue it all together seamlessly. Our problem as artists is to get some of this down on a piece of flat paper. So let’s try find out what’s really going on and how we might use it.


spherical perspective, drawing

So if we take our camera and take tall thin pictures of an endless gridded floor and see what we get. I’m using a virtual one as endless gridded floors are a bit thin on the ground around here. The image above is the result  in which everything joins up neatly. It looks pretty fisheye when it gets very close to us, but every paving stone joins every other where it should and is the right size for its distance from us. It’s also plain that It will repeat all the way round 360 degrees. That’s good too because it’s a well known fact that endless gridded floors do just that!


spherical, perspective, drawing

Here is the same method applied to our polkadot floor which looked so weird in the first example. This is a 180 degree view so very wide, but none of the ellipses are tilted and everything joins up in a logical manner. The problem of ellipses in perspective is a very old one which renaissance artists puzzled over a fair bit. This was due to their often needing to draw long rows of receding cylindrical columns which using linear perspective would look distinctly wonky on the far left and right. They devised a simple cheat which modern artists seem to have forgotten, but I will deal with that in the next instalment.


cylindrical, perspective, spherical, drawing

Here is our grid joined up so you can see how verticals work, you could easily imagine laying out a cityscape on this grid. You can repeat it endlessly if you mirror it left or right. This is the most useful curved perspective and is called Cylindrical perspective it is the equivalent of two point perspective, I will deal with the horrors of full Spherical perspective next! But first below an example of Cylindrical perspective by the wonderful M C Esher.


Escher, perspective, cylindrical

This is called House of Stairs and is made using the same grid I drew above, but swivelled through 90 degrees.


Spherical, perspective, drawing

Now we are entering the strange world of Spherical perspective. The above grid can as before be duplicated endlessly, I know it looks like an impossible spiders web but it is the same as the cylindrical one except the verticals curve too. This is the equivalent of 3 point perspective as when you look up at a tall building and the top diminishes. The image above can be clicked for a larger version.


spherical, perspective, drawing

Here is the same grid used to place a few very simple forms. I takes a little while to get your head around it but if you print out the grid and scribble on top of it you will soon get the idea. For the mathematically minded the geometry of the Spherical grid is Hyperbolic whereas the traditional straight line perspective is termed Euclidian.


spherical, perspective, drawing

Here it is used in anger. You can click for a larger view. As with all spherical perspective the nearer you get to looking down at your toes the odder it looks!


spherical, perspective, drawing, panorama

Here it is finished. You can click for a larger view. The examples I have given are extreme ones in order to show the principles involved. We don’t often draw or paint 180 degree panoramas. But the same principles can be applied to good effect in more ordinary views where to the casual viewer the underlying curves would be to subtle to notice but they will unconsciously find the painting just that little bit more believable especially in townscapes and other subjects with a lot of man made rectilinear content.


  1. WOW! Thanks for this OUTSTANDING tutorial. I have never seen anything like it anywhere. Bravo!

    Comment by Jan — April 17, 2011 @ 1:26 pm

  2. Great Tutorial! Thank You!

    Comment by John — May 31, 2011 @ 12:27 am

  3. Wow Excellent explanation. I got so many answers in this site.Thanks.

    Comment by madhu kuruva — October 28, 2011 @ 8:00 am

  4. This is a beautifully illustrated exposition of some commonly held concepts on curvilinear perspective, but unfortunately it is based on a fundamental misunderstanding. The concept of linear perspective is to project the world onto a flat canvas (or film plane). Regular lenses do this without distortion, but if you swing the camera around to capture a wide angle scene, you are moving the film plane to a different orientation for each shot, so the compound image that you stitch together does not correspond to the projection to a single picture plane. It is therefore an interesting form of compound perspective corresponding to a continuous view over time, not true linear perspective. It may be taken as a cognitive representation of the unified experience of the natural perspective of an observer in the world, but it is a distortion from the single-view geometry.

    Comment by Christopher Tyler — December 5, 2011 @ 12:28 am

  5. Thanks for comment Christopher, I’m not sure where you think I am “misunderstanding”
    “The concept of linear perspective is to project the world onto a flat canvas (or film plane)”
    Yes, but the film plane does not have to be flat. It can for example be cylindrical (as many first cameras were) and then the film unrolled flat, so no view over time.
    Or a spherical film surface is used and then Mercator’s method of flattening the spherical surface to a flat surface applied. Both of these are projections on to a flat surface so that is something all projection methods share.
    The actual geometry of our vision is Hyperbolic rather than the single perspectives Euclidian view, both can be expressed on a 2 dimensional surface.
    “Regular lenses do this without distortion.”
    This is entirely untrue. If you read my post on columns you will see what I mean. Photographic images are all distorted to some degree though we are so used to the distortions that we correct unconsciously.

    Comment by admin — December 5, 2011 @ 9:38 am

  6. hey, a nice read. Just wanted to share my take on spherical perspective, or what is it, 4 point perspective? 360×180 degrees drawing of some hands. click and drag to interact with a drawing. scroll to zoom in / out. for extra maddens click left mouse button and choose little planet view.

    Comment by mantas — May 28, 2013 @ 11:27 am

  7. I don’t usually allow links but this one is fine, fun drawing. Spherical persp has 6 vanishing points if fully spherical, 4 if cylindrical. Most panoramas are cylindrical. The different view choices on the viewer use those different projections.

    Comment by admin — May 28, 2013 @ 12:57 pm

  8. Thanks for going through all this effort to share this knowledge with us!

    Comment by Zachary Madere — June 25, 2013 @ 6:57 pm

  9. Lindos desenhos em perspectiva aqui publicados. parabéns!

    Comment by yvonne tessuto tavares — August 3, 2013 @ 12:30 pm

  10. Thank you for your presentation. I had been arguing with a painter, my friend, about ellipses. He insists that they must be “perfect”. With a real center, equidistant, which axes intersect, major and minor. But I thought a fitted ellipse within a square had to share its center, and thus would draw with that distirsión. Like when I draw the bow of a building between two columns. I struggled to understand that the laws of perspective are not completely faithful criteria. (Sorry for my limited english). Thank you again.

    Comment by Ricardo — October 17, 2013 @ 8:21 pm

  11. Hi Ricardo. A perspective ellipse is distorted but only the interior construction. See below the ellipse has not become distorted because as the angle changes the points where the axes meet the circle move around the ellipse. See also this post: Columns

    Comment by Rob Adams — October 18, 2013 @ 9:57 am

  12. I had already carefully examined the graphs to which you direct me . And it was enough to understand the problem. I also see the solution you propose to fit the columns , in the square of the ground. Columns win. Squares lost. I’ve had to draw the colonnades of monasteries , which are cut in half ellipses and square structures embedded in , and I wonder if the next time I have to resolve with the method ” arches win, lose squares ” . Besides, I also have to disguise the lack of consistency between the two structures . Did I understand right?

    Masters of the past have sacrificed reality for the benefit of aesthetics. For example , a critical analysis of the painting ” Miss Lola au Cirque Fernando ” by Degas , London, National Gallery- , stated, that verticality of the columns of the circus were false . Should tend to converge on a vanishing point . But Degas chose a solution , aesthetics , not real.

    These challenges are the spice of life . Do not ? Kind regards .

    Comment by Ricardo — October 18, 2013 @ 6:36 pm

  13. Part of the problem is that people think photographs and linear perspective are “real” and accurate. They are not, all methods of getting a 3 dimensional world on to a two dimensional bit of paper are a projection of one kind or another which always means there are distortions. In the same way as flat maps of the world can never really show the Continents correctly, only a globe can do that.

    Comment by Rob Adams — October 18, 2013 @ 7:50 pm

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